topic entry on axioms and foundations of homology and cohomology theories
1 Axioms for Homology and Cohomology theories

1.
Axioms for homology^{} theory and uniqueness theorems

2.
Cech types

3.
$K$theory

4.
Generalized cohomology

5.
Generalized (extraordinary) homology and cohomology^{} theories

6.
Galois Cohomology and Categorical Galois theories

7.
(Co)homology of commutative rings and algebras (e.g., Hochschild, André–Quillen, cyclic, dihedral, etc.)

8.
Bordism^{} and cobordism theories, formal group laws

9.
Homology with local coefficients, equivariant cohomology
 10.

11.
Cohomology in Noncommutative algebraic geometry

12.
Classifying spaces^{} for foliations; GelfandFuks cohomology

13.
Intersection homology and cohomology

14.
Elliptic cohomology

15.
Equivariant homology and cohomology

16.
Homology and homotopy^{} of topological groups^{} and related structures

17.
Homotopy Quantum Field Theories and Axiomatic Quantum Field Theories

18.
NonAbelian^{} Homological Algebra

19.
Grothendieck’s ‘Anabelian Geometry’

20.
Other homology theories–Your new additions
References
 1 Hatcher, A. 2001. http://www.math.cornell.edu/ hatcher/AT/AT.pdfAlgebraic Topology (textbook on line)., Cambridge University Press; Cambridge, UK., 405 pages.
 2 Ronald Brown: Topology and Groupoids, BookSurge LLC (2006).
 3 Ronald Brown R, P.J. Higgins, and R. Sivera.: “NonAbelian algebraic topology”, (2008).
 4 R. Brown and J.L. Loday: Homotopical excision, and Hurewicz theorems, for ncubes of spaces, Proc. London Math. Soc., 54:(3), 176192, (1987).
 5 R. Brown and J.L. Loday: Van Kampen Theorems^{} for diagrams of spaces, Topology, 26: 311337 (1987).
 6 R. Brown and C.B. Spencer: Double groupoids^{} and crossed modules, Cahiers Top. Géom. Diff., 17 (1976), 343362.
 7 Allain Connes: Noncommutative Geometry, Academic Press 1994.
 8 May, J.P. 1999, A Concise Course in Algebraic Topology., The University of Chicago Press: Chicago
Title  topic entry on axioms and foundations of homology and cohomology theories 
Canonical name  TopicEntryOnAxiomsAndFoundationsOfHomologyAndCohomologyTheories 
Date of creation  20130322 18:20:43 
Last modified on  20130322 18:20:43 
Owner  bci1 (20947) 
Last modified by  bci1 (20947) 
Numerical id  12 
Author  bci1 (20947) 
Entry type  Topic 
Classification  msc 11E72 
Classification  msc 55N99 
Classification  msc 55N25 
Classification  msc 55N30 
Classification  msc 55N05 
Classification  msc 12G05 
Classification  msc 14A22 
Classification  msc 11R34 
Classification  msc 55N40 
Synonym  fundamentals of homology and cohomology theories 